|
%I #50 Jan 06 2023 15:31:43
%S 0,1,1,1,4,10,28,85,262,829,2677,8776,29143,97825,331381,1131409,
%T 3889381,13450744,46764532,163357807,573064849,2018027719,7131064045,
%U 25278463756,89866690732,320328538033,1144591699069,4099050204445,14710315969696,52893571881142
%N Number of planar planted trees with n non-root nodes and every 2-valent node isolated.
%C Isolated 2-valent node is a 2-valent node non-adjacent to any other 2-valent node.
%C Hankel transform of a(n+1) is A174108. Binomial transform of a(n+1) is A174107. - _Paul Barry_, Mar 07 2010
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (Problem 2.7.4).
%H Vincenzo Librandi, <a href="/A061639/b061639.txt">Table of n, a(n) for n = 0..1000</a>
%H Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, <a href="http://jl.baril.u-bourgogne.fr/bacorasisito.pdf">Consecutive pattern-avoidance in Catalan words according to the last symbol</a>, Univ. Bourgogne (France 2023).
%F a(n) = Sum_{m=0..n-1} Sum_{i=0..n-m-1} (-1)^i/(m+1)*binomial(2*m, m)*binomial(m+i, i)*binomial(m+i+1, n-m-i-1).
%F G.f.: (1/2)*(1-sqrt(1-4*(x+x^2)/(1+x+x^2))).
%F From _Paul Barry_, Mar 07 2010: (Start)
%F G.f.: (x(1+x)/(1+x+x^2))c(x(1+x)/(1+x+x^2)), c(x) the g.f. of A000108;
%F G.f.: (1+x+x^2-sqrt(1-2x-5x^2-6x^3-3x^4))/(2*(1+x+x^2)). (End)
%F Conjecture: n*a(n) +2*(2-n)*a(n-1) +(14-5*n)*a(n-2) +6*(3-n)*a(n-3) +3*(4-n)*a(n-4)=0. - _R. J. Mathar_, Nov 15 2011
%F a(n) ~ 7^(1/4) * 2^(n-7/2) * 3^(n+1/4) / (sqrt(Pi) * n^(3/2) * (sqrt(21)-3)^(n-1/2)). - _Vaclav Kotesovec_, Feb 12 2014
%F From _Peter Bala_, May 30 2017: (Start)
%F G.f. A(x) satisfies the differential equation (3*x^4 + 6*x^3 + 5*x^2 + 2*x - 1)*A(x)' - (4*x + 2)*A(x) + 2*x + 1 = 0 with A(0) = 0. Mathar's conjectural recurrence above follows from this.
%F 1 - A(x) + A(x)^2 = 1/(1 + x + x^2). (End)
%F From _Nikolaos Pantelidis_, Apr 11 2022: (Start)
%F G.f.: f(x) = (sqrt(1+x+x^2)-sqrt(1-3*x-3*x^2))/(2*sqrt(1+x+x^2)).
%F Series reversion of g.f. f(x) is -f(-x). (End) [Corrected by _Joerg Arndt_, Jun 08 2022]
%t CoefficientList[Series[1/2*(1-Sqrt[1-4*(x+x^2)/(1+x+x^2)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%K nonn,easy
%O 0,5
%A _Vladeta Jovovic_, Jun 13 2001
| 